arXiv subjects

Robert de Mello Koch

Publications and source records attributed to Robert de Mello Koch.

At least 109 records · Page 6Linked to original sources

arXiv2011

Nonplanar Integrability: Beyond the SU(2) Sector

We compute the one loop anomalous dimensions of restricted Schur polynomials with a classical dimension Δ\sim O(N). The operators that we consider are labeled by Young diagrams with two long columns or two long rows. Simple analytic expressions for the action of the dilatation operator are found. The projection operators needed to define the restricted Schur polynomials are constructed by translating the problem into a spin chain language, generalizing earlier results obtained in the SU(2) sector of the theory. The diagonalization of the dilatation operator reduces to solving five term recursion relations. The recursion relations can be solved exactly in terms of products of symmetric Kravchuk polynomials with Hahn polynomials. This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable, extending a similar conclusion reached for the SU(2) sector of the theory.

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arXiv2011

Nonplanar Integrability

In this article we study operators with a dimension $Δ\sim O(N)$ and show that simple analytic expressions for the action of the dilatation operator can be found. The operators we consider are restricted Schur polynomials. There are two distinct classes of operators that we consider: operators labeled by Young diagrams with two long columns or two long rows. The main complication in working with restricted Schur polynomials is in building a projector from a given $S_{n+m}$ irreducible representation to an $S_n\times S_m$ irreducible representation (both specified by the labels of the restricted Schur polynomial). We give an explicit construction of these projectors by reducing it to the simple problem of addition of angular momentum in ordinary non-relativistic quantum mechanics. The diagonalizationof the dilatation operator reduces to solving three term recursion relations. The fact that the recursion relations have only three terms is a direct consequence of the weak mixing at one loop of the restricted Schur polynomials. The recursion relations can be solved exactly in terms of symmetric Kravchuk polynomials or in terms of Clebsch-Gordan coefficients. This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable.

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arXiv2010

Surprisingly Simple Spectra

The large N limit of the anomalous dimensions of operators in ${\cal N}=4$ super Yang-Mills theory described by restricted Schur polynomials, are studied. We focus on operators labeled by Young diagrams that have two columns (both long) so that the classical dimension of these operators is O(N). At large N these two column operators mix with each other but are decoupled from operators with $n\ne 2$ columns. The planar approximation does not capture the large N dynamics. For operators built with 2, 3 or 4 impurities the dilatation operator is explicitly evaluated. In all three cases, in a certain limit, the dilatation operator is a lattice version of a second derivative, with the lattice emerging from the Young diagram itself. The one loop dilatation operator is diagonalized numerically. All eigenvalues are an integer multiple of $8g_{YM}^2$ and there are interesting degeneracies in the spectrum. The spectrum we obtain for the one loop anomalous dimension operator is reproduced by a collection of harmonic oscillators. This equivalence to harmonic oscillators generalizes giant graviton results known for the BPS sector and further implies that the Hamiltonian defined by the one loop large $N$ dilatation operator is integrable. This is an example of an integrable dilatation operator, obtained by summing both planar and non-planar diagrams.

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arXiv2010

AdS_4/CFT_3 Construction from Collective Fields

We pursue the construction of higher-spin theory in AdS_4 from CFT_3 of the O(N) vector model in terms of canonical collective fields. In null plane quantization an exact map is established between the two spaces. The coordinates of the AdS_4 space-time are generated from the collective coordinates of the bi-local field. This, in the light cone gauge, provides an exact one to one reconstruction of bulk AdS_4 space-time and higher-spin fields.

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arXiv2010

Emergent Threebrane Lattices

In this article the anomalous dimension of a class of operators with a bare dimension of O(N) is studied. The operators considered are dual to excited states of a two giant graviton system. In the Yang Mills theory they are described by restricted Schur polynomials, labeled with Young diagrams that have at most two columns. In a certain limit the dilatation operator looks like a lattice version of a second derivative, with the lattice emerging from the Young diagram itself.

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arXiv2010

From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group

We show that correlators of the hermitian one-Matrix model with a general potential can be mapped to the counting of certain triples of permutations and hence to counting of holomorphic maps from world-sheet to sphere target with three branch points on the target. This allows the use of old matrix model results to derive new explicit formulae for a class of Hurwitz numbers. Holomorphic maps with three branch points are related, by Belyi's theorem, to curves and maps defined over algebraic numbers $\bmQ$. This shows that the string theory dual of the one-matrix model at generic couplings has worldsheets defined over the algebraic numbers and a target space $ \mP^1 (\bmQ)$. The absolute Galois group $ Gal (\bmQ / \mQ) $ acts on the Feynman diagrams of the 1-matrix model, which are related to Grothendieck's Dessins d'Enfants. Correlators of multi-matrix models are mapped to the counting of triples of permutations subject to equivalences defined by subgroups of the permutation groups. This is related to colorings of the edges of the Grothendieck Dessins. The colored-edge Dessins are useful as a tool for describing some known invariants of the $ Gal (\bmQ / \mQ) $ action on Grothendieck Dessins and for defining new invariants.

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arXiv2009

Emergent Spacetime

We give an introductory account of the AdS/CFT correspondence in the 1/2-BPS sector of ${\cal N}=4$ super Yang-Mills theory.Six of the dimensions of the string theory are emergent in the Yang-Mills theory. In this article we suggest how these dimensions and local physics in these dimensions emerge. The discussion is aimed at non-experts.

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arXiv2009

Hints of Integrability Beyond the Planar Limit: Nontrivial Backgrounds

The problem of computing the anomalous dimensions of a class of (nearly) half-BPS operators with a large R-charge is reduced to the problem of diagonalizing a Cuntz oscillator chain. Due to the large dimension of the operators we consider, non-planar corrections must be summed to correctly construct the Cuntz oscillator dynamics. These non-planar corrections do not represent quantum corrections in the dual gravitational theory, but rather, they account for the backreaction from the heavy operator whose dimension we study. Non-planar corrections accounting for quantum corrections seem to spoil integrability, in general. It is interesting to ask if non-planar corrections that account for the backreaction also spoil integrability. We find a limit in which our Cuntz chain continues to admit extra conserved charges suggesting that integrability might survive.

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arXiv2009

Correlators Of Operators with a Large R-charge

Correlation functions of operators with a conformal dimension of O(N^2) are not well approximated by the planar limit. The non-planar diagrams, which in the bulk spacetime correspond to string loop corrections, are enhanced by huge combinatorial factors. In this article we show how these loop corrections can be resummed. As a typical example of our results, in the half-BPS background of M maximal giant gravitons we find the usual 1/N expansion is replaced by a 1/(M+N) expansion. Further, we find that there is a simple exact relationship between amplitudes computed in the trivial background and amplitudes computed in the background of M maximal giant gravitons. Finally, we also find strong evidence for the BMN-type sectors suggested in arXiv:0801.4457. The decoupling limit of arXiv:0801.4457 captures the decoupled low energy world volume theory of the intersecting giant graviton system and this theory is weakly coupled even when the original N=4 super Yang-Mills theory is strongly coupled.

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arXiv2009

Correlators in Nontrivial Backgrounds

Operators in N=4 super Yang-Mills theory with an R-charge of O(N^2) are dual to backgrounds which are asymtotically AdS5xS5. In this article we develop efficient techniques that allow the computation of correlation functions in these backgrounds. We find that (i) contractions between fields in the string words and fields in the operator creating the background are the field theory accounting of the new geometry, (ii) correlation functions of probes in these backgrounds are given by the free field theory contractions but with rescaled propagators and (iii) in these backgrounds there are no open string excitations with their special end point interactions; we have only closed string excitations.

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arXiv2008

Geometries from Young Diagrams

Type IIB string theory on spacetimes that are asymptotically AdS$_5\times$S$^5$ can be defined using four dimensional ${\cal N}=4$ super Yang-Mills theory. Six of the dimensions of the string theory are holographically reconstructed in the Yang-Mills theory. In this article we study how these dimensions and local physics in these dimensions emerge. We reorganize the dynamics of the ${1\over 2}$ BPS sector of the field theory by rewriting it in terms of Schur polynomials. The Young diagram labeling of these polynomials can be viewed as a book keeping device which summarizes how the operator is constructed. We show that aspects of the geometry of the extra holographic dimensions are captured very naturally by the Young diagram. Gravitons which are localized at a specific position in the bulk correspond to boxes added at a specific location on the Young diagram.

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arXiv2008

Exact Multi-Restricted Schur Polynomial Correlators

We derive a product rule satisfied by restricted Schur polynomials. We focus mostly on the case that the restricted Schur polynomial is built using two matrices, although our analysis easily extends to more than two matrices. This product rule allows us to compute exact multi-point correlation functions of restricted Schur polynomials, in the free field theory limit. As an example of the use of our formulas, we compute two point functions of certain single trace operators built using two matrices and three point functions of certain restricted Schur polynomials, exactly, in the free field theory limit. Our results suggest that gravitons become strongly coupled at sufficiently high energy, while the restricted Schur polynomials for totally antisymmetric representations remain weakly interacting at these energies. This is in perfect accord with the half-BPS (single matrix) results of hep-th/0512312. Finally, by studying the interaction of two restricted Schur polynomials we suggest a physical interpretation for the labels of the restricted Schur polynomial: the composite operator $χ_{R,(r_n,r_m)}(Z,X)$ is constructed from the half BPS ``partons'' $χ_{r_n}(Z)$ and $χ_{r_m}(X)$.

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arXiv2008

Exact Multi-Matrix Correlators

We argue that restricted Schur polynomials provide a useful parameterization of the complete set of gauge invariant variables of multi-matrix models. The two point functions of restricted Schur polynomials are evaluated exactly in the free field theory limit. They have diagonal two point functions.

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arXiv2008

Matrix Model Maps and Reconstruction of AdS SUGRA Interactions

We consider the question of reconstructing (cubic) SUGRA interactions in AdS/CFT. The method we introduce is based on the matrix model maps (MMP) which were previously successfully employed at the linearized level. The strategy is to start with the map for 1/2 BPS configurations which is exactly known (to all orders) in the hamiltonian framework. We then use the extension of the matrix model map with the corresponding Ward identities to completely specify the interaction. A central point in this construction is the non-vanishing of off-shell interactions (even for highest-weight states).

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arXiv2008

Giant Gravitons - with Strings Attached (III)

We develop techniques to compute the one-loop anomalous dimensions of operators in the ${\cal N}=4$ super Yang-Mills theory that are dual to open strings ending on boundstates of sphere giant gravitons. Our results, which are applicable to excitations involving an arbitrary number of open strings, generalize the single string results of hep-th/0701067. The open strings we consider carry angular momentum on an S$^3$ embedded in the S$^5$ of the AdS$_5\times$S$^5$ background. The problem of computing the one loop anomalous dimensions is replaced with the problem of diagonalizing an interacting Cuntz oscillator Hamiltonian. Our Cuntz oscillator dynamics illustrates how the Chan-Paton factors for open strings propagating on multiple branes can arise dynamically.

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arXiv2007

Giant Gravitons - with Strings Attached (II)

We study the one-loop anomalous dimensions of operators in the ${\cal N}=4$ super Yang-Mills theory that are dual to open strings ending on giant gravitons. We consider both AdS and sphere giants as well as boundstates of them. The open strings we consider carry angular momentum on an S^3 embedded in the S^5 of the AdS$_5\times$S^5 background. The main result of this article is that we derive a bosonic lattice Hamiltonian that describes the one loop mixing of the operators dual to the general excited giant graviton system. A semiclassical analysis of the Hamiltonian allows us to give a geometrical interpretation for the labeling used to describe the gauge theory operators. We also argue that AdS giant gravitons are unstable against the excitations considered.

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arXiv2007

Giant Gravitons - with Strings Attached (I)

In this article, the free field theory limit of operators dual to giant gravitons with open strings attached are studied. We introduce a graphical notation, which employs Young diagrams, for these operators. The computation of two point correlation functions is reduced to the application of three simple rules, written as graphical operations performed on the Young diagram labels of the operators. Using this technology, we have studied gravitational radiation by giant gravitons and bound states of giant gravitons, transitions between excited giant graviton states and joining of open strings attached to the giant.

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arXiv2006

Unstable Giants

We find giant graviton solutions in Frolov's three parameter generalization of the Lunin-Maldacena background. The background we study has $\tildeγ_1=0$ and $\tildeγ_2=\tildeγ_3=\tildeγ$. This class of backgrounds provide a non-superymmetric example of the gauge theory/gravity correspondence that can be tested quantitatively, as recently shown by Frolov, Roiban and Tseytlin. The giant graviton solutions we find have a greater energy than the point gravitons, making them unstable states. Despite this, we find striking quantitative agreement between the gauge theory and gravity descriptions of open strings attached to the giant.

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